Linear programming problems with two decision variables can be easily
solved by **graphical method**. A problem of three dimensions can also be
solved by this method, but their graphical solution becomes complicated.

Don't misconstrue that the graphical method is very important. It provides very little help in handling practical problems.

## Basic Terminology - Graphical Method

### Feasible Region

It is the collection of all feasible solutions. In the following figure,
the shaded area represents the feasible region.

### Convex Set

A region or a set R is convex, if for any two points on the set R,
the segment connecting those points lies entirely in R. In other words,
it is a collection of points such that for any two points on the set,
the line joining the points belongs to the set. In the following figure,
the line joining P and Q belongs entirely in R.

Thus, the collection of feasible solutions in a linear
programming problem form a convex set.

### Redundant
Constraint

It is a constraint that does not affect the feasible region.

Example: Consider the linear programming problem:

Maximize 1170 x_{1 }+ 1110x_{2}

subject to

9x_{1 }+ 5x_{2} ≥ 500

7x_{1 }+ 9x_{2} ≥ 300

5x_{1 }+ 3x_{2} ≤ 1500

7x_{1 }+ 9x_{2} ≤ 1900

2x_{1 }+ 4x_{2} ≤ 1000

x_{1}, x_{2} ≥ 0

The feasible region is indicated in the following figure:

The critical region has been formed by the two constraints.

9x_{1 }+ 5x_{2} ≥ 500

7x_{1 }+ 9x_{2} ≤ 1900

x_{1}, x_{2} ≥ 0

The remaining three constraints are not affecting the feasible region
in any manner. Such constraints are called **redundant constraints**.

### Extreme Point

Extreme points are referred to as vertices or corner points. In the
following figure, P, Q, R and S are extreme points.